Die design: An implicit formulation for the inverse problem

Let us call a direct extrusion problem (DEP) the problem of finding the shape of the extrudate coming out of a die of prescribed shape. An implicit finite element formulation of the DEP which is geometrically general and for which a Newton-Raphson technique can be implemented has recently been proposed by Legat and Marchal. However, the problem posed to the die designer is frequently the inverse extrusion problem (IEP), i.e. finding the die shape which produces an extrudate of prescribed shape. This paper presents an extension of our original method for solving the IEP which avoids the ‘trial-and-error’ iteration on the die geometry itself. The advantage of the formulation lies in its capability to handle complex geometrics and in its low cost, because the CPU time and memory required to solve the IEP are almost identical to those of the DEP. We present benchmark results for squares and rectangles and new results obtained for geometries involving multiple corners. For an octagonal shape we also consider the case of a power-law fluid. For all results presented in this paper, surface tension has not been included.